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Transactions of the American Mathematical Society

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Compact traveling waves for anisotropic curvature flows with driving force


Authors: H. Monobe and H. Ninomiya
Journal: Trans. Amer. Math. Soc. 374 (2021), 2447-2477
MSC (2020): Primary 35C07; Secondary 34-XX
DOI: https://doi.org/10.1090/tran/8168
Published electronically: January 20, 2021
MathSciNet review: 4223022
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Abstract: To study the dynamics of an anisotropic curvature flow with external driving force depending only on the normal vector, we focus on traveling waves composed of Jordan curves in $\mathbb {R}^2$. Here we call them compact traveling waves. The objective of this study is to investigate thoroughly the condition of the driving force for the existence of compact traveling waves to the anisotropic curvature flow. It is shown that all traveling waves are strictly convex and unstable, and that a compact traveling wave is unique, if they exist. To determine the existence of compact traveling waves, three cases are considered: if the driving force is positive, there exists a compact traveling wave; if it is negative, there is no traveling wave; if it is sign-changing, a positive answer is obtained under the assumption called “admissible condition”. We also obtain a necessary and sufficient condition for the existence of axisymmetric compact traveling waves. Lastly, we make reference to the inverse problem and non-convex compact traveling waves.


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Additional Information

H. Monobe
Affiliation: Research Institute for Interdisciplinary Science, Okayama University, 3-1-1 Tsushi ma-naka, Kita-ku, Okayama, 700-8530, Japan
MR Author ID: 937404
Email: monobe@okayama-u.ac.jp

H. Ninomiya
Affiliation: School of Interdisciplinary Mathematical Sciences, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo 164-8525, Japan
MR Author ID: 330408
ORCID: 0000-0001-7081-6564

Keywords: Differential geometry, algebraic geometry
Received by editor(s): July 27, 2019
Received by editor(s) in revised form: March 9, 2020
Published electronically: January 20, 2021
Additional Notes: The first author was supported in part by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number 15K17595.
The second author was supported in part by JSPS KAKENHI Grant Numbers 26287024, 15K04963, 16K13778, 16KT0022.
Article copyright: © Copyright 2021 American Mathematical Society